D O C U M E N T 4 1 8 C A L C U L A T I O N S I N D I A R Y 6 9 1 by , symmetrize it with respect to the indices σ and τ, and set it equal to 0. The plus signs in front of the two bracketed terms in the second line should be minus signs, and the minus sign in front of the term in the second line (which appears to have been corrected from a plus sign) should be a plus sign. With the correct signs, the resulting equation is equivalent to eq. (10) of Doc. 417. [15] and in [eq. 6] should be and , respectively. Einstein then lowers the indi- ces σ and τ of [eq. 6] by multiplying with , then writes down the resulting equation three times with cyclically permutated indices , subtracting the first equation and adding the second and third equations to obtain [eq. 7]. [16]Einstein here introduces the notation . This use of a comma in separating the indices of the connection symbols is also made in Weyl 1921a, p. 113, eqs. (49) and (50), as well as in the form in Eddington 1921a, p. 109, eq. (4.2). See also note 22. [17]For the following, see Eddington 1921a, pp. 108–109. [18]For a version of the compatibility condition for metric and connection, see Doc. 417, p. 1. [19]See similar calculations on the spherically symmetric case on pp. [44] and [45]. [20]See [eq. 2] and note 13. [21]Perhaps an allusion to the same difficulties referred to in his diary entry of 13 January (Doc. 379). [22]Einstein at this point implicitly introduces the notation of using semicolons to separate sub- script indices for covariant derivatives. The use of a semicolon to denote a covariant derivation was introduced by Einstein earlier in Einstein 1922c, p. 46, eq. (71), and p. 47, eq. (78) (Vol. 7, Doc. 71, pp. 545 and 546, respectively). Neither Weyl 1921a nor Eddington 1921b used a semicolon in their tensorial notations (see note 16 above). Palatini 1919b used a vertical bar in the subscript to denote covariant derivatives, a notation that he had introduced in Palatini 1919a, p. 198, from eq. (12) on. [23]Einstein apparently considered, for a brief moment, a variational principle of the form (he omitted the ), but may have discarded it immediately because he did not want to go down the route of quadratic field equations. He then changed the action integral to His new choice, using what appears to be the determinant of the matrix , as written here, is actually ambiguous because it is unclear whether he takes the new integrand in the action to be or . The choice of the determinant of the for the Hamiltonian indicates a departure from the draft manuscript (Doc. 417) and a step toward the approach of the published ver- sion, Einstein 1923e (Doc. 425). At this point Einstein appears to take the determinant only over the symmetric components of . [24]Einstein apparently introduced the notation to rewrite the action integral as . Again, Einstein omitted the in his variational integral. The calculations on the remainder of the page are an explicit calculation of this expression. For a similar calculation, see below on [p. 47] and [p. 46v]. [25]Einstein inserted [eq. 9] into [eq. 8] and performed partial integrations in the first two terms to obtain a variational integral written here in two lines. The expression is incomplete, the integral lacks the closing bracket and a factor dτ, and it should be set equal to 0. Introducing factors of and rela- beling indices, Einstein proceeded to write down in the next line the integrand A of an expression of –g gαβΓαβ σ Tσiκ Tτiκ Γσiκ Γτiκ gσαgτβ αβμ βμα μαβ →→ Γβ, αμ gτβΓαμ τ ≡ Γσμ, ν gανΓσμ α ≡ δ RiκRiκdτ 0= dτ δ Riκ dτ 0. = Riκ Riκ Riκ Riκ Rik Rik Riκ Riκ Riκ ≡ δ RiκRiκdτ δ Riκ 2 Riκ ------------------dτ 1 2 -- - Riκ Riκ δRiκ Riκ ------------------------------ -dτ 1 2 -- - Riκ Riκ δRiκdτ 1 2 -- - RiκδRiκdτ = = = = dτ δα κ